Question

given right triangle def, what is the value of tan(f)?
options:
$\frac{9}{41}$
$\frac{40}{41}$
$\frac{40}{9}$
$\frac{41}{9}$

Explanation:

Step1: Recall the definition of tangent in a right triangle

In a right triangle, the tangent of an acute angle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \( F \) in right triangle \( DEF \) (right - angled at \( D \)):

• The side opposite to angle \( F \) is \( DE \) with length \( 40 \).
• The side adjacent to angle \( F \) is \( DF \) with length \( 9 \).

Step2: Calculate \( \tan(F) \)

Using the formula for tangent of an angle in a right triangle, \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). For \( \theta = F \), we have:
\( \tan(F)=\frac{DE}{DF}=\frac{40}{9} \)

Answer:

\( \boldsymbol{\frac{40}{9}} \) (corresponding to the option with \( \frac{40}{9} \))